Furthermore, the definition of security for encrypted control systems is ambiguous, and the types of attacks against the systems are not clearly defined.
Meanwhile, the security of cryptosystems has been mathematically defined and analyzed in the field of cryptography [1]}.
One of the security approaches, called provable security, is formulated based on the objectives and capabilities of the attackers.
Additionally, it is proved under some assumptions of computational complexity.
More precisely, the security is usually demonstrated using proof by contradiction, namely if there exists an algorithm to break a cryptosystem, then a practically difficult computational problem can be efficiently solved by using the algorithm.
This proof construction is called the reductionist approach and is fundamental in proving the security of modern cryptosystems.
The security of encrypted control systems can be clarified and formally analyzed by applying a similar security definition and proof construction.
Consequently, the standard security definition of cryptosystems must be expanded to encompass the specific attacks for encrypted control systems because the attack scenarios considered in cryptography and control systems vary significantly.
Some studies have already applied cryptographic security to encrypted control protocols.
The privacy of cloud-based optimization, model predictive control, and state estimation is achieved through the computational indistinguishability of two random ensembles [2]}, [3]}, [4]}, and that of distributed and cooperative controls is achieved based on a cryptographic game [5]}, [6]}.
However, the guaranteed privacy of the aforementioned studies is based on the input/output privacy of the systems rather than a controller parameter and is provided for specific protocols.
